# The Number Behind the Simplest SIC–POVM

@article{Bengtsson2016TheNB, title={The Number Behind the Simplest SIC–POVM}, author={Ingemar Bengtsson}, journal={Foundations of Physics}, year={2016}, volume={47}, pages={1031-1041} }

The simple concept of a SIC poses a very deep problem in algebraic number theory, as soon as the dimension of Hilbert space exceeds three. A detailed description of the simplest possible example is given.

## 24 Citations

SICs and Algebraic Number Theory

- Mathematics, Physics
- 2017

We give an overview of some remarkable connections between symmetric informationally complete measurements (SIC-POVMs, or SICs) and algebraic number theory, in particular, a connection with Hilbert’s…

Sporadic SICs and the Exceptional Lie Algebras

- Physics, Mathematics
- 2019

Sometimes, mathematical oddities crowd in upon one another, and the exceptions to one classification scheme reveal themselves as fellow-travelers with the exceptions to a quite different taxonomy.

Fibonacci Fervour in Linear Algebra and Quantum Information Theory

- Mathematics, Physics
- 2018

This is a survey on certain results which bring about a connection between Fibonacci sequences on the one hand and the areas of matrix theory and quantum information theory, on the other.

Moment maps and Galois orbits for SIC-POVMs

- Physics, Mathematics
- 2019

The equations that define covariant SIC-POVMs are interpreted in terms of moment maps. Attention is focussed on orbits of a cyclic subgroup of a maximal torus and their images in the moment polytope.…

Invariant Off-Diagonality: SICs as Equicoherent Quantum States

- Physics
- 2019

Coherence, treated as a resource in quantum information theory, is a basis-dependent quantity. Looking for states that have constant coherence under canonical changes of basis yields highly symmetric…

Fibonacci-Lucas SIC-POVMs

- Mathematics, Physics
- 2017

We present a conjectured family of symmetric informationally complete positive operator valued measures which have an additional symmetry group whose size is growing with the dimension. The symmetry…

Quantum Theory as Symmetry Broken by Vitality

- Physics, Mathematics
- 2019

I summarize a research program that aims to reconstruct quantum theory from a fundamental physical principle that, while a quantum system has no intrinsic hidden variables, it can be understood using…

Modular Welch Bounds with Applications

- Mathematics
- 2022

Results (1) and (2) reduce to the famous result of Welch [IEEE Transactions on Information Theory, 1974 ] obtained 48 years ago. We introduce the notions of modular frame potential, modular…

The Poincaré Half-Plane for Informationally-Complete POVMs

- Computer Science, MathematicsEntropy
- 2018

The structure of some IC-POVMs is found to be intimately related to the Kochen–Specker theorem.

The SIC Question: History and State of Play

- Computer Science, MathematicsAxioms
- 2017

Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844.

## References

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- Mathematics, Physics
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We give an overview of some remarkable connections between symmetric informationally complete measurements (SIC-POVMs, or SICs) and algebraic number theory, in particular, a connection with Hilbert’s…

Sporadic SICs and the Normed Division Algebras

- Mathematics, Physics
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Symmetric informationally complete quantum measurements, or SICs, are mathematically intriguing structures, which in practice have turned out to exhibit even more symmetry than their definition…

SIC-POVMs: A new computer study

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We report on a new computer study into the existence of d2 equiangular lines in d complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite dimensions and are…

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The finite Heisenberg group knows when the dimension of Hilbert space is a square number. Remarkably, it then admits a representation such that the entire Clifford group—the automorphism group of the…

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It is conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.

Galois automorphisms of a symmetric measurement

- Mathematics, PhysicsQuantum Inf. Comput.
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The Galois group of SICs covariant with respect to the Weyl-Heisenberg group is examined and a list of nine conjectures concerning its structure are proposed, representing a considerable strengthening of the theorems actually proved.

Symmetric informationally complete positive-operator-valued measures: A new computer study

- Mathematics
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We report on a new computer study of the existence of d2 equiangular lines in d complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite dimensions and are the…

Gaussian and covariant processes in discrete and continuous variable quantum information

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Quantum information theory has attracted much interest in the last decade. The cause of this interest is twofold: the exciting applications that the theory promises, such as the realization of…

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The existence of maximal sets of equiangular lines (SIC-POVMs) is of interest to the mathematics and physics communities due to their connection to quantum information theory, quantum cryptography,…

SIC‐POVMS and MUBS: Geometrical Relationships in Prime Dimension

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The paper concerns Weyl‐Heisenberg covariant SIC‐POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When…